What are the pros and cons of studying machine learning before deep learning?
That question doesn't really make sense: deep learning is a sub-topic of machine learning, so you can't really 'skip' it. It's a bit like "I want to learn about trigonometry, but do I need to do geometry first?"

Having said that, in order to make sense of deep learning you should really know about the general principles of machine learning, otherwise you won't understand it. Or, more importantly, you won't understand what problems deep learning can be applied to, and what issues are better solved with other methods.

You don't need to go into much detail, but should at least get an overview.